Square Numbers. Oblong Numbers. ─────────────┐ ───────────────┐ • • •  │ • • • • │ ───────┐ │ ───────────┐ │ • • │ • │ • • • │ • │ ──┐ │ │ ──────┐ │ │ • │ • │ • │ • • │ • │ • │ It is clear, then, that we are entitled to refer the study of sums of series to Pythagoras himself; but whether he went beyond the oblong, and studied pyramidal or cubic numbers, we cannot say.[239] Footnote 238: Cf. Milhaud, _Philosophes géomètres_, pp. 115 sqq. Aristotle puts the matter thus (_Phys._ Γ, 4. 203 a 13): περιτιθεμένων γὰρ τῶν γνωμόνων περὶ τὸ ἓν καὶ χωρὶς ὁτὲ μὲν ἄλλο ἀεὶ γίγνεσθαι τὸ εἶδος, ὁτὲ δὲ ἕν. This is more clearly stated by Ps.-Plut. (Stob. i. p. 22, 16), Ἔτι δὲ τῇ μονάδι τῶν ἐφεξῆς περισσῶν περιτιθεμένων ὁ γινόμενος ἀεὶ τετράγωνός ἐστι· τῶν δὲ ἀρτίων ὁμοίως περιτιθεμένων ἑτερομήκεις καὶ ἄνισοι πάντες ἀποβαίνουσιν, ἴσως δὲ ἰσάκις οὐδείς. I cannot feel satisfied with any of the explanations which have been given of the words καὶ χωρίς in the Aristotelian passage (see Zeller, p. 351, n. 2), and I would therefore suggest ταῖς χώραις comparing Boutheros (Stob. i. p. 19, 9), who says, according to the MS. reading, Καὶ ὁ μὲν (ὁ περισσός), ὁπόταν γεννῶνται ἀνὰ λόγον καὶ πρὸς μονάδας, ταῖς αὑτοῦ χώραις καταλαμβάνει τοὺς ταῖς γραμμαῖς περιεχομένους (sc. ἀριθμούς). Footnote 239: In the fragment referred to above (p. 113, _n._ 236), Speusippos speaks of four as the first pyramidal number; but this is taken from Philolaos, so we cannot safely ascribe it to Pythagoras. [Sidenote: Geometry and harmonics.]